Questions tagged [complex-manifolds]

For questions about complex manifolds.

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75 views

Geometric Wedge Products of Two Currents and Push Forward a Current by an Automorphism of $\mathbb{P}^2$

Harmonic Currents of Finite Energy and Laminations, J. E. Fornaess and N. Sibony, GAFA, Geom, Funct, Anal, $2005$, $962-1003$. Page $993$. Let $\mathbb{P}^2$ be the Complex Projective Space and $$\...
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64 views

$\nabla\omega=0$ if and only if $(M,g)$ is Kähler

Given an almost complex manifold $(M,g)$ with an almost complex structure $J$, we know that $(M,g)$ is Kähler if and only if $J$ is integrable and $\mathrm{d}\omega=0$, where $\omega$ is the Kähler ...
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2answers
59 views

Existence of $f\colon U\to V$ holomorphic non constant

Let $Y$ be a complex mainfold and $\Omega\subseteq\Bbb C^n$ domain, $n<\dim Y$. Take $z_0\in\Omega$ and $y_0\in Y$. I need a non-constant holomorphic mapping $f\colon U\to V$ where $U\subset Y$ and ...
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51 views

Hodge decomposition for complex manifold

Let $X$ be a compact oriented Riemannian manifold. By Hodge decomposition, we can decompose $$\Omega^k(X)=\mathrm{im}(d)\oplus\mathrm{im}(d^*)\oplus\ker(\Delta).$$ Now, if further $X$ has a complex ...
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47 views

Interchange of limits of differential operators on differential forms

Let $M$ be a complex 2-dimensional manifold with the additional property that a smooth function $h:M \to \mathbb C$ is constant if and only if $\partial\overline\partial h = 0$ (which is equivalent (...
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2answers
76 views

Chern classes and sums of line bundles

Let $E$ be a complex vector bundle of rank $r$ and suppose we can write $E = \oplus_{i=1}^r L_i$ where $L_i$ are line bundles. I have read here (and think I more or less understand why) that the total ...
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27 views

Representation Theorem for Laminated Positive Harmonic Currents on Laminations with Singularities

Harmonic Currents of Finite Energy and Laminations, J. E. Fornaess and N. Sibony, GAFA, Geom, Funct, Anal, $2005$, $962-1003$. Page $977$, Theorem $3.5$. Any other highly recommended references to ...
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16 views

Laminated Positive Harmonic Currents on a Compact Complex Manifold as an Integration aginst a Positive Borel Measure on a Transversal

Riemann Surface Laminations with Singularities, J. E. Fornaess and N. Sibony, J. From Anal, $2008$, $400-442$. Page $414$, Definition $7$. Let $\theta$ be a $(1,1)$ form on $M$. How can the formula, ...
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24 views

Is there any theorem that says "a nonlinear system of equations with less equations rather than unknowns has infinite complex solutions"?

I am working on a paper and I am wondering if there's any mathematical theorem that says a system of nonlinear equations has infinite many complex (or real) solutions when there are m equations and n ...
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35 views

Exact, Closed and Harmonic Currents on Compact Complex Manifolds and their Quotient Spaces

Let $M$ be a compact complex manifold of complex dimension $m$. It is well-known that, since every exact current is closed, we can define $$D_{current}^p(M)=\frac{\{Closed p-currents\}}{\{Exact p-...
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49 views

Doubt about proof Proper Mapping Theorem (Remmert)

I am studying the proof of Remmert's Theorem on the book Griffiths & Harris - Principles of Algebraic Geometry, Chapter 3 Section 2 page 395. Theorem (Remmert's Proper Mapping Theorem) Let $U$ and ...
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45 views

Regarding the Definition of Harmonic Current on a Compact Complex Manifold

Harmonic Currents of Finite Energy and Laminations, J. E. Fornaess and N. Sibony, Geometric And Functional Analysis, $2005$, Page $965$, Section $2$ "Harmonic Currents". Definition $2.1$ Let ...
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1answer
49 views

Non-elementary Fuchsian Group and the Invariant Measure for the Corresponding Action on the Limit Set

Let $\Gamma$ be a Fuchsian Group (that means a Discrete Subgroup of $RSL_2(\mathbb{R})$) acting on the Closed Unit Disc $\mathbb{D}$. Let $\Lambda (\Gamma)$ be the Limit Set of $\Gamma$, i.e., it is ...
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1answer
40 views

Restrictions on coordinate basis of $T_pM$ required for a manifold to admit a Hermitian metric?

I asked this question about relating the Riemannian metric on a manifold $M$ to the Hermitian metric that arises when $M$ is thought of as a complex manifold (i.e. with integrable complex structure). ...
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1answer
34 views

Growth of Leaves of a Foliation on a Manifold

Let $M$ be a (Real or Complex) Manifold of dimension $m$ and $\mathcal{F}$ a Foliation of dimension $k$, $1 \leq k \leq m-1$, on $M$. I am actually looking for highly recommended references on the ...
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30 views

Is the complexification $M^{\mathbb{C}}$ of a complex manifold $M$ fibered over $M$?

Denote by $\mathcal{O}_{\cdot,p}$ the $\mathbb{C}$-algebra of a complex manifold at a given point $p$. Let $M$ be a complex manifold and $M^{\mathbb{C}}$ the complexification of the associated real ...
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1answer
67 views

How to Prove a Riemann Surface is Hyperbolic by Showing its Volume Grows Exponentially?

We have a Holomorphic Foliation by curves on a Complex Manifold. Therefore, every leaf is a Riemann Surface. Let $R$ be a Riemann Surface. We say that the Riemann Surface $R$ is Hyperbolic if and only ...
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1answer
36 views

Complex Manifold question

So let's take the following: $$C\subset \mathbb{P}^2(\mathbb{C})$$ where $$C = \left\{[x_0:x_1:x_2]; x_0^2 + x_1^2 + x_2^2 = 0 \right\}$$ How can I proceed to prove that $C$ is a complex manifold? ($...
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1answer
33 views

How do you construct a complex manifold?

I am attending courses about complex manifolds and the teacher gave us the property to construct complex manifolds, that follows : " Let X be a complex manifold and $\Gamma \subset Aut( \textbf{X}...
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1answer
89 views

The Closed and $\partial \overline{\partial} -$ Closed Current

Let $M$ be a Complex Manifold of complex dimension $m$. Let $T$ be a Current on $M$. What does Directed Current mean? The current $T$ is said to be Closed if $dT=0$. What does The Current $T$ is $\...
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1answer
73 views

Currents on Complex Manifolds

The concept of Current is definitely a well-known one in Functional Analysis, Manifolds,.. etc. I am actually looking for highly recommended references on Currents on Complex Manifolds. Thank in ...
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40 views

A Real Analytic Variety $N$ of Real Dimension $n$ of a Complex Manifold $M$ of Complex Dimension $m$

Let $M$ be a Complex Manifold of complex dimension $m$. A Real Analytic Variety of $M$ is a closed subset $N \subseteq M$ such that for any point $p\in N$ there exists an open neighbourhood $p\in U \...
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34 views

Linearisability of Holomorphic Vector Fields and Poincaré-Dulac Normal Form

Complex Algebraic Foliations, Book by A. Lins Neto and Bruno Scárdua, Chapter 1, Page 36. The last two lines, Don't you think they should have written that the vector field is locally conjugate (...
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1answer
53 views

The Multiplier of a Biholomorphism of the Complex Projective Space at a Fixed Point

Let $Aut(\mathbb{C} \mathbb{P}(1))$ be the Automorphism Group of $\mathbb{C} \mathbb{P}(1)$ ($\mathbb{C} \mathbb{P}(1)=$ the Complex Projective Space of dimension one). Let $T \in Aut(\mathbb{C} \...
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33 views

The Right way of identifying in the Blow-Up of $\mathbb{C}^2$ at the Origin $(0,0)\in \mathbb{C}^2$

It is well-known that the Blow-Up of $\mathbb{C}^2$ at the Origin $(0,0)$ is a Complex Manifold $\mathbb{\tilde{C}}^2$ obtained by identifying two copies of $\mathbb{C}^2$. Which one is the right way ...
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17 views

The Index of a Singular Point relative to a Singular One-Dimensional Holomorphic Foliation on a Complex Surface

Let $S$ be a Complex Surface, $C$ a Compact Non-Singular Curve and $\mathcal{F}$ a Singular One-Dimensional Holomorphic Foliation on the complex surface $S$ leaving the curve $C$ Invariant. Let $p\in ...
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43 views

How is The Number of Self-Intersections of a Compact Connected Riemann Surface Embedded in a Complex Surface defined?

Complex Algebraic Foliations, Book by A. Lins Neto and Bruno Scárdua, Chapter 3, Page 82. Let $M$ be a Complex Manifold of complex dimension two (i.e., $M$ is a Complex Surface). Let $S$ be a Compact ...
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19 views

The Eigenvalues of a Singular One-Dimensional Holomorphic Foliation at a Singular Point

Let $M$ be a Complex Manifold of complex dimension $n$. Let $\mathcal{F}$ be a Singular One-Dimensional Holomorphic Foliation on the complex manifold $M$. Let $p \in M$ be a Singular Point of the ...
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1answer
39 views

Regarding the Definition of Holomorphic Foliation on a Complex Manifold

Geometry, Dynamics And Topology Of Foliations: A First Course, Book by Bruno Scárdua and Carlos Arnoldo Morales Rojas, Chapter 1, Page 33. Definition 1.14. The definition of Holomorphic Foliation on a ...
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1answer
82 views

Proving that a (complex) differential form is of type $(p, q)$ iff its conjugate is of type $(q, p)$ using complex vector fields

Let $(M, J)$ be a $n$-complex manifold and $p+q=k$ where $0\leq k \leq 2n$. I'm looking for a clean way to prove that $\mu\in \mathcal{A}^k(M, \mathbb{C}):=\Gamma\left(\bigwedge^k T_{\mathbb C}^*M \...
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1answer
167 views

Relatively minimal elliptic surfaces which are not minimal

A complex surface is called minimal if it contains no $(-1)$-curves, while an elliptic surface $\pi : X \to C$ is called relatively minimal if the fibers of $\pi$ contain no $(-1)$-curves. It follows ...
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53 views

Why do people consider multivalued solutions of differential equations?

The nature of the question is heuristic. If we have, say, a nonsingular complex manifold, what is the interest in knowing solutions of a differential equation of its universal cover instead of on the ...
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35 views

Canonical isomorphism from $\Lambda^{(p,q)} (V \otimes \mathbb{C})$ to $\Lambda^{p+q} (V \otimes \mathbb{C})$

I'm having trouble understanding this 'natural' isomorphism when discussing complex differential forms. Let $(M, J)$ be an almost complex manifold. Then its complexified cotangent bundle decomposes as ...
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27 views

Quotient of a nonconstant meromorphic function on a hyperelliptic curve

Let $X$ be a hyperelliptic curve, and let $f:X\to \Bbb P^1$ be the degree 2 map induced by hyperelliptic involution. By elementary properties of holomorphic maps, $f$ is surjective, continuous, and ...
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1answer
77 views

Problem book recommendations on complex manifolds

I came across the book on Cauchy Riemann manifolds, "CR manifolds and tangential Cauchy Riemann complexes". The book does not have a problem section. I would be grateful if anyone recommends ...
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1answer
74 views

Total space of $\mathscr{O}(1)$

In this Wikipedia article I read: This [the canonical quotient map $\mathbf{C}^{n+1} \setminus \{0\} \twoheadrightarrow \mathbf{CP}^n = (\mathbf{C}^{n+1}\setminus \{0\}) / \mathbf{C}^\times$] ...
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11 views

Conjugate on complex manifold with respect to a fixed coordinate chart.

I am interested in a kind of 'conjugate' on a complex manifold with respect to a fixed coordinate chart. Let $M$ be a $1$-diminsional complex manifold and $(U_\lambda,\varphi_\lambda)_{\lambda\in\...
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1answer
125 views

When $\mathbb{C}/\Lambda$ is a Riemann surface?

Let $\Lambda$ be a lattice, that is $\Lambda=\{a\omega_1+b\omega_2\mid a,b\in\mathbb Z\}$. I heard that the necessarily and sufficient condition that $\mathbb{C}/\Lambda$ be a Riemann surface is $\...
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1answer
122 views

Relation between symplectic manifolds and (almost) complex manifolds

I'm a beginner in symplectic geometry, and I recently learned that every symplectic manifold has an almost complex structure. I am curious about the converse. Does every almost complex manifold have a ...
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1answer
61 views

On what open subset of $\mathbb C$ can derivation be surjective?

We define the sheaf of holomorphic functions on $\mathbb C$, hence for any holomorphic function on an open subset we can do differentiation. I want to ask on what kind of open subset $U$ can it be ...
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Why is this map into the Lewy hypersurface a bijection?

Let $\mathbb{S}^{2N - 1}$ be the unit sphere in $\mathbb{C}^N$ and let $L$ be the Lewy hypersurface, which is defined by $$ \operatorname{Im} Z_N = \sum_{j = 0}^{N - 1} |Z_j|^2. $$ Let $H(\mathbf{Z}):\...
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1answer
127 views

Holomorphic and Harmonic $1-$ Forms on Riemann Surfaces

Let $\mathbb{S}$ be a Riemann Surface. We say that a differential form $\omega$, on the Riemann Surface $\mathbb{S}$, is a Harmonic $1-$ Form if it and its conjugate $\omega^\star$ are both closed, i....
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45 views

Easier proof that the Grassmannian is a complex manifold

$G_r(\mathbb C^3,2)$ is the topological space of 2-dimensional complex linear subspaces of $\mathbb C^3$. Prove that $G_r(\mathbb C^3,2)$ is a complex manifold. I have a solution to this problem, but ...
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129 views

Questions about algebraic analysis: prerequesites, references and state of the field [closed]

I'm interested in studying algebraic analysis, specifically the area described in here https://en.wikipedia.org/wiki/Algebraic_analysis and also the theory of D-modules, as I find the idea of "...
3
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1answer
159 views

Holomorphic function on a connected compact Riemann surface is constant

I was trying to solve the following exercise. I wanted to check if my solution was correct/rigorous enough, and ask a question at the end. (The general direction is given in here: holomorphic map ...
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63 views

Existence of coframe for Hermitian metric on complex manifold?

I am reading page 28 of the 1994 version of Principles of Algebraic Geometry by Griffith. Let $M$ be a complex manifold of dimension n, Griffith defined a Hermitian metric to be a positive definite ...
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47 views

Connected components of the isotropic Grassmannian

Let $W$ be a $2n$-dimensional complex vector space endowed with a non-degenerate, symmetric, bilinear form $Q$. We choose Euclidean coordinates on $W$ such that $Q$ is represented by symmetric ...
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23 views

Conditions on $X,\Omega$ such that $g(K)$ is a Stein compactum in $X$ $\forall g:\Omega\to X$ holomorphic and $\forall K\Subset\Omega$.

Let $\Omega\subset\Bbb C^n$ open bounded and $X$ complex manifold. I am searching for some condition on $X,\Omega$ such that $g(K)$ is a Stein compactum in $X$ for every $g:\Omega\to X$ holomorphic ...
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1answer
43 views

Any complex curve in a complex surface is the zero set of some holomorphic section of a holomorphic line bundle

Let $Y$ be a closed complex surface, $L\to Y$ be a holomorphic line bundle, $\sigma:Y\to L$ a holomorphic section, and $B\subset Y$ the zero set of $\sigma$. If the first Chern class $c_1(L)$ of $L$ ...
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133 views

On the complexification of a holomorphic bundle $E$

Let $E\to M$ be a holomorphic vector bundle over a complex manifold. Considering it as a real vector bundle one can complexify $E$ to $E^{\mathbb{C}}$. The bundle $E^{\mathbb{C}}$ is again holomorphic ...

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